Moduli spaces of meromorphic connections and the Fourier transform
Final Report Abstract
In this project, we studied topics around differential systems with irregular singularities, which play an important role in different parts of mathematics and physics. A major focus was on the investigation of a fundamental operation: the Fourier transformation. Meromorphic differential equations on a Riemann surface can be characterized by linear-algebraic data, their so-called Stokes data, and these are the basis for constructing moduli spaces of such objects. Together with Jean Douçot and building on results of Takuro Mochizuki and Philip Boalch, we were able to describe an explicit algorithmic way to compute the Stokes data of the Fourier transform starting from the Stokes data of the original system in a large class of cases. On the one hand, this adds new examples to the quite restricted list of explicit cases understood so far. On the other hand, this concrete algorithm enabled us to verify in these cases that the Fourier transform is compatible with certain geometric structures on the moduli spaces. Another focus of our project was on the use of rather recent techniques developed by Andrea D’Agnolo and Masaki Kashiwara for the study of irregular singularities: the theory of enhanced ind-sheaves. First, we investigated a question of descent: Concretely, we were interested in finding conditions under which systems with irregular singularities – and hence in particular their Stokes data – are defined over certain subfields of the complex numbers. In a joint work with Davide Barco, Marco Hien and Christian Sevenheck, we gave an explicit answer to this question in the case of hypergeometric differential equations. Subsequently, further questions on conjugation and descent for sheaves and enhanced ind-sheaves were investigated. To answer them, we developed in particular some complements on compatibilities of operations on constructible sheaves in a joint work with Pierre Schapira. Furthermore, Kashiwara’s conjugation functor for D-modules was investigated in the context of the results of D’Agnolo–Kashiwara. Another aspect was studied in a joint work with Brian Hepler, where we showed how to recover classical objects describing solutions with growth conditions in the framework of enhanced ind-sheaves. In order to do this, we established in particular a duality between the De Rham complexes with moderate growth and rapid decay, which closely relates to some classical duality results due to Bloch–Esnault and Hien. This also opens up perspectives on studying nearby and vanishing cycles for irregular D-modules with these new methods.
Publications
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Betti Structures of Hypergeometric Equations. International Mathematics Research Notices, 2023(12), 10641-10701.
Barco, Davide; Hien, Marco; Hohl, Andreas & Sevenheck, Christian
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The notion of R-triple and various functors, Extended Abstract, Oberwolfach Report No. 17/2023, 12–14.
A. Hohl
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Kashiwara conjugation and the enhanced Riemann–Hilbert correspondence. Portugaliae Mathematica, 81(3), 347-387.
Hohl, Andreas
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Moderate Growth and Rapid Decay NearbyCycles via Enhanced Ind-Sheaves. Publications of the Research Institute for Mathematical Sciences, 61(1), 1-51.
Hepler, Brian & Hohl, Andreas
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Unusual functorialities for weakly constructible sheaves. Pacific Journal of Mathematics, 334(1), 153-166.
Hohl, Andreas & Schapira, Pierre
